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Theorem wpthswwlks2on 28948
Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.)
Assertion
Ref Expression
wpthswwlks2on ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (𝐴(2 WSPathsNOn 𝐺)𝐡) = (𝐴(2 WWalksNOn 𝐺)𝐡))

Proof of Theorem wpthswwlks2on
Dummy variables 𝑓 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlknon 28844 . . . . . . 7 (𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡))
21a1i 11 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)))
32anbi1d 631 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ ((𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ ((𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
4 3anass 1096 . . . . . . 7 ((𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)))
54anbi1i 625 . . . . . 6 (((𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ ((𝑀 ∈ (2 WWalksN 𝐺) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
6 anass 470 . . . . . 6 (((𝑀 ∈ (2 WWalksN 𝐺) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
75, 6bitri 275 . . . . 5 (((𝑀 ∈ (2 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
83, 7bitrdi 287 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ ((𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ (𝑀 ∈ (2 WWalksN 𝐺) ∧ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))))
98rabbidva2 3412 . . 3 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ {𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = {𝑀 ∈ (2 WWalksN 𝐺) ∣ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)})
10 usgrupgr 28175 . . . . . . . . . . 11 (𝐺 ∈ USGraph β†’ 𝐺 ∈ UPGraph)
11 wlklnwwlknupgr 28873 . . . . . . . . . . 11 (𝐺 ∈ UPGraph β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) ↔ 𝑀 ∈ (2 WWalksN 𝐺)))
1210, 11syl 17 . . . . . . . . . 10 (𝐺 ∈ USGraph β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) ↔ 𝑀 ∈ (2 WWalksN 𝐺)))
1312bicomd 222 . . . . . . . . 9 (𝐺 ∈ USGraph β†’ (𝑀 ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)))
1413adantr 482 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (𝑀 ∈ (2 WWalksN 𝐺) ↔ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)))
15 simprl 770 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ 𝑓(Walksβ€˜πΊ)𝑀)
16 simprl 770 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) β†’ (π‘€β€˜0) = 𝐴)
1716adantr 482 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (π‘€β€˜0) = 𝐴)
18 fveq2 6847 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘“) = 2 β†’ (π‘€β€˜(β™―β€˜π‘“)) = (π‘€β€˜2))
1918ad2antll 728 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (π‘€β€˜(β™―β€˜π‘“)) = (π‘€β€˜2))
20 simprr 772 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) β†’ (π‘€β€˜2) = 𝐡)
2120adantr 482 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (π‘€β€˜2) = 𝐡)
2219, 21eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (π‘€β€˜(β™―β€˜π‘“)) = 𝐡)
23 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2423wlkp 28606 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walksβ€˜πΊ)𝑀 β†’ 𝑀:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ))
25 oveq2 7370 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘“) = 2 β†’ (0...(β™―β€˜π‘“)) = (0...2))
2625feq2d 6659 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘“) = 2 β†’ (𝑀:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ↔ 𝑀:(0...2)⟢(Vtxβ€˜πΊ)))
2724, 26syl5ibcom 244 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walksβ€˜πΊ)𝑀 β†’ ((β™―β€˜π‘“) = 2 β†’ 𝑀:(0...2)⟢(Vtxβ€˜πΊ)))
2827imp 408 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ 𝑀:(0...2)⟢(Vtxβ€˜πΊ))
29 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ 𝑀:(0...2)⟢(Vtxβ€˜πΊ))
30 2nn0 12437 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ β„•0
31 0elfz 13545 . . . . . . . . . . . . . . . . . . . . . 22 (2 ∈ β„•0 β†’ 0 ∈ (0...2))
3230, 31mp1i 13 . . . . . . . . . . . . . . . . . . . . 21 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ 0 ∈ (0...2))
3329, 32ffvelcdmd 7041 . . . . . . . . . . . . . . . . . . . 20 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ (π‘€β€˜0) ∈ (Vtxβ€˜πΊ))
34 nn0fz0 13546 . . . . . . . . . . . . . . . . . . . . . . 23 (2 ∈ β„•0 ↔ 2 ∈ (0...2))
3530, 34mpbi 229 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ (0...2)
3635a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ 2 ∈ (0...2))
3729, 36ffvelcdmd 7041 . . . . . . . . . . . . . . . . . . . 20 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ (π‘€β€˜2) ∈ (Vtxβ€˜πΊ))
3833, 37jca 513 . . . . . . . . . . . . . . . . . . 19 (𝑀:(0...2)⟢(Vtxβ€˜πΊ) β†’ ((π‘€β€˜0) ∈ (Vtxβ€˜πΊ) ∧ (π‘€β€˜2) ∈ (Vtxβ€˜πΊ)))
3928, 38syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ ((π‘€β€˜0) ∈ (Vtxβ€˜πΊ) ∧ (π‘€β€˜2) ∈ (Vtxβ€˜πΊ)))
40 eleq1 2826 . . . . . . . . . . . . . . . . . . 19 ((π‘€β€˜0) = 𝐴 β†’ ((π‘€β€˜0) ∈ (Vtxβ€˜πΊ) ↔ 𝐴 ∈ (Vtxβ€˜πΊ)))
41 eleq1 2826 . . . . . . . . . . . . . . . . . . 19 ((π‘€β€˜2) = 𝐡 β†’ ((π‘€β€˜2) ∈ (Vtxβ€˜πΊ) ↔ 𝐡 ∈ (Vtxβ€˜πΊ)))
4240, 41bi2anan9 638 . . . . . . . . . . . . . . . . . 18 (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ (((π‘€β€˜0) ∈ (Vtxβ€˜πΊ) ∧ (π‘€β€˜2) ∈ (Vtxβ€˜πΊ)) ↔ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))))
4339, 42imbitrid 243 . . . . . . . . . . . . . . . . 17 (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ ((𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))))
4443adantl 483 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) β†’ ((𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))))
4544imp 408 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))
46 vex 3452 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
47 vex 3452 . . . . . . . . . . . . . . . 16 𝑀 ∈ V
4846, 47pm3.2i 472 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑀 ∈ V)
4923iswlkon 28647 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝑓 ∈ V ∧ 𝑀 ∈ V)) β†’ (𝑓(𝐴(WalksOnβ€˜πΊ)𝐡)𝑀 ↔ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜(β™―β€˜π‘“)) = 𝐡)))
5045, 48, 49sylancl 587 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (𝑓(𝐴(WalksOnβ€˜πΊ)𝐡)𝑀 ↔ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜(β™―β€˜π‘“)) = 𝐡)))
5115, 17, 22, 50mpbir3and 1343 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ 𝑓(𝐴(WalksOnβ€˜πΊ)𝐡)𝑀)
52 simplll 774 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ 𝐺 ∈ USGraph)
53 simprr 772 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (β™―β€˜π‘“) = 2)
54 simpllr 775 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ 𝐴 β‰  𝐡)
55 usgr2wlkspth 28749 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (β™―β€˜π‘“) = 2 ∧ 𝐴 β‰  𝐡) β†’ (𝑓(𝐴(WalksOnβ€˜πΊ)𝐡)𝑀 ↔ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
5652, 53, 54, 55syl3anc 1372 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ (𝑓(𝐴(WalksOnβ€˜πΊ)𝐡)𝑀 ↔ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
5751, 56mpbid 231 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) ∧ (𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2)) β†’ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)
5857ex 414 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) β†’ ((𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
5958eximdv 1921 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
6059ex 414 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
6160com23 86 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑀 ∧ (β™―β€˜π‘“) = 2) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
6214, 61sylbid 239 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (𝑀 ∈ (2 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
6362imp 408 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ 𝑀 ∈ (2 WWalksN 𝐺)) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) β†’ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀))
6463pm4.71d 563 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ 𝑀 ∈ (2 WWalksN 𝐺)) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ↔ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)))
6564bicomd 222 . . . 4 (((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) ∧ 𝑀 ∈ (2 WWalksN 𝐺)) β†’ ((((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀) ↔ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)))
6665rabbidva 3417 . . 3 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ {𝑀 ∈ (2 WWalksN 𝐺) ∣ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀)} = {𝑀 ∈ (2 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)})
679, 66eqtrd 2777 . 2 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ {𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀} = {𝑀 ∈ (2 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)})
6823iswspthsnon 28843 . 2 (𝐴(2 WSPathsNOn 𝐺)𝐡) = {𝑀 ∈ (𝐴(2 WWalksNOn 𝐺)𝐡) ∣ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑀}
6923iswwlksnon 28840 . 2 (𝐴(2 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (2 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜2) = 𝐡)}
7067, 68, 693eqtr4g 2802 1 ((𝐺 ∈ USGraph ∧ 𝐴 β‰  𝐡) β†’ (𝐴(2 WSPathsNOn 𝐺)𝐡) = (𝐴(2 WWalksNOn 𝐺)𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  {crab 3410  Vcvv 3448   class class class wbr 5110  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  2c2 12215  β„•0cn0 12420  ...cfz 13431  β™―chash 14237  Vtxcvtx 27989  UPGraphcupgr 28073  USGraphcusgr 28142  Walkscwlks 28586  WalksOncwlkson 28587  SPathsOncspthson 28705   WWalksN cwwlksn 28813   WWalksNOn cwwlksnon 28814   WSPathsNOn cwwspthsnon 28816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-concat 14466  df-s1 14491  df-s2 14744  df-s3 14745  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-wlks 28589  df-wlkson 28590  df-trls 28682  df-trlson 28683  df-pths 28706  df-spths 28707  df-pthson 28708  df-spthson 28709  df-wwlks 28817  df-wwlksn 28818  df-wwlksnon 28819  df-wspthsnon 28821
This theorem is referenced by:  usgr2wspthons3  28951  frgr2wsp1  29316
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